Traci here on Monday night -- Above is my attempt to insert the completed Vignettes (Word doc) so everyone can see. I added one more vignette, because I had to add another row when I split up the flu stories, and that one little flu story looked awfully lonely in a row by itself.
I (Stacey) think that in order to show a situation where deductive reasoning is used, we don't necessarily have to give the proof . If we are doing a non-mathematical example, a "proof" could be impossible, so we could just clearly depict the use of deductive reasoning - relying on definitions, axioms (commonly accepted notions?), theorems (accepted academic theories in other fields like sociology, physics, economics, statistics, . . . ?).
A mathematical story of deduction:
While doing her homework one night, Sahara found herself factoring a lot of numbers. She noticed that every time she factored an odd number, she got only odd factors. She saw that 17 = 17(1 ), 25 = 5(5), 51 = 3(17). Sahara wondered if this would always happen.
"What do I know about odd numbers? I know that odd numbers can be expressed in the general form 2s + 1, and I also know that they will all be either prime or composite. Prime will be simple. If p is prime and odd, the factors of p are p and 1. p is known to be odd, and 1 is odd, so the factors of a prime odd number will always be prime.
"Odd composites will be a bit harder. How can I do this? What do I know from the definition of composite numbers? . . ."
Stacey
Vignette using deductive reasoning:
All of the students who earned a B or higher on Ms. Jackson's mathematics exam skipped school today. Jenna, one of Ms. Jackson's students, attended school today. Therefore, Jenna earned a C or below on Ms. Jackson's exam.
Bill Thill
Another one:
If the measure of an angle is 39.7 degrees, and I know that itis complimentary to another angle, then what must be true about the second angle? By definition, I know that a complimentary angle means . . . so I can find some things that must be true by solving . . .
(Marty) Taken together and compared to the inductive examples I posted:
1) The rule f=2d will calculate the number of people with the flu, f, given the number of days, d, from the start of the outbreak. Therefore, after 3 days, six people will have the flu.
2) The rule f=2^d will calculate the number of people with the flu, f, given the number of days, d, from the start of the outbreak. Therefore, after 3 days, eight people will have the flu.
And another one:
Joe is 16 years old and Lesley is 4 years younger than Joe. However, Javier is two times as old as Lesley. How old is LeighAnn if she is 6 years younger than Xavier?
Marcela saw a webpage that showed the official portraits (pictures) of all the Presidents of the United States. When she looked at President Kennedy's picture, she thought he looked really young. For a few seconds, she wondered if he might have even been younger than her uncle, who is 34. But then she remembered that there is a minimum age for people to be elected President. She looked in her American Government book, and found out that the Constitution says someone must be at least 35 years old to be elected president. So she decided President Kennedy must have been older than her uncle at the time of his official portrait.
Traci here on Monday night -- Above is my attempt to insert the completed Vignettes (Word doc) so everyone can see. I added one more vignette, because I had to add another row when I split up the flu stories, and that one little flu story looked awfully lonely in a row by itself.
I (Stacey) think that in order to show a situation where deductive reasoning is used, we don't necessarily have to give the proof . If we are doing a non-mathematical example, a "proof" could be impossible, so we could just clearly depict the use of deductive reasoning - relying on definitions, axioms (commonly accepted notions?), theorems (accepted academic theories in other fields like sociology, physics, economics, statistics, . . . ?).
A mathematical story of deduction:
While doing her homework one night, Sahara found herself factoring a lot of numbers. She noticed that every time she factored an odd number, she got only odd factors. She saw that 17 = 17(1 ), 25 = 5(5), 51 = 3(17). Sahara wondered if this would always happen.
"What do I know about odd numbers? I know that odd numbers can be expressed in the general form 2s + 1, and I also know that they will all be either prime or composite. Prime will be simple. If p is prime and odd, the factors of p are p and 1. p is known to be odd, and 1 is odd, so the factors of a prime odd number will always be prime.
"Odd composites will be a bit harder. How can I do this? What do I know from the definition of composite numbers? . . ."
Stacey
Vignette using deductive reasoning:
All of the students who earned a B or higher on Ms. Jackson's mathematics exam skipped school today. Jenna, one of Ms. Jackson's students, attended school today. Therefore, Jenna earned a C or below on Ms. Jackson's exam.
Bill Thill
Another one:
If the measure of an angle is 39.7 degrees, and I know that it is complimentary to another angle , then what must be true about the second angle ? By definition, I know that a complimentary angle means . . . so I can find some things that must be true by solving . . .
(Marty) Taken together and compared to the inductive examples I posted:
1) The rule f=2d will calculate the number of people with the flu, f, given the number of days, d, from the start of the outbreak. Therefore, after 3 days, six people will have the flu.
2) The rule f=2^d will calculate the number of people with the flu, f, given the number of days, d, from the start of the outbreak. Therefore, after 3 days, eight people will have the flu.
And another one:
Joe is 16 years old and Lesley is 4 years younger than Joe. However, Javier is two times as old as Lesley. How old is LeighAnn if she is 6 years younger than Xavier?
Marcela saw a webpage that showed the official portraits (pictures) of all the Presidents of the United States. When she looked at President Kennedy's picture, she thought he looked really young. For a few seconds, she wondered if he might have even been younger than her uncle, who is 34. But then she remembered that there is a minimum age for people to be elected President. She looked in her American Government book, and found out that the Constitution says someone must be at least 35 years old to be elected president. So she decided President Kennedy must have been older than her uncle at the time of his official portrait.